Neighborhood Insurance sells fire insurance policies to local homeowners. The premium is $270, the probability of a fire is 0.1%, and in the event of a fire, the insured damages (the payout on the policy) will be $260,000.

a. Make a table of the two possible payouts on each policy with the probability of each.
b. Suppose you own the entire firm, and the company issues only one policy. What are the expected value, variance and standard deviation of your profit?
c. Now suppose your company issues two policies. The risk of fire is independent across the two policies. Make a table of the three possible payouts along with their associated probabilities. (Round your "Probability" answers to 4 decimal places.)
d. What are the expected value, variance and standard deviation of your profit?
e. Compare your answers to (b) and (d). Did risk pooling increase or decrease the variance of your profit?
f. Continue to assume the company has issued two policies, but now assume you take on a partner, so that you each own one-half of the firm. Make a table of your share of the possible payouts the company may have to make on the two policies, along with their associated probabilities. (Round your "Probability" answers to 4 decimal places.)
g. What are the expected value and variance of your profit?

Explanation:

There are two possible payout scenarios - 1) There is a fire 2) There is no fire. So payout table will look like below:

Scenario 1Scenario 2

FireNo-Fire

Payout260,0000

Probability0.10%99.90%

Answer B)

Expected Value of profit = Profit from scenario 1 * probability of scenario 1 + Profit from scenario 2 * probability of scenario 2

Profit from scenario 1 = Premium Collected - Insurance Payout = 270 - 260,000 = -259730

Profit from scenario 2 = Premium Collected

Scenario 1Scenario 2

FireNo-Fire

Profit-259730270given

Probability0.10%99.90%given

Profit*Probability-259.73269.7310=Expected Value

Variance

= (Profit from scenario 1)^2 * probability of scenario 1 + (Profit from scenario 2)^2 * probability of scenario 2 - (Expected Value)^2

Standard Deviation = Square root of variance

Scenario 1Scenario 2

VarianceFireNo-Fire

Profit^2 67,459,672,900.00 72,900.00

Probability0.10%99.90%

Profit^2 * Probability 67,459,672.90 72,827.10 67,532,400.00=Variance

8,217.81=Standard Deviation

So, the expected income is $10, with a variance of $8,217.81

Answer C)

Joint Probability of two independent events = probability of event 1*probability of event 2

ScenarioProbability

Single Case - No fire99.90%

Single Case - fire0.10%

ScenarioJoin ProbabilityPayouts

No Fire99.8001%0

One Fire0.0999% 260,000

Two Fire0.0001% 520,000

Answer D)

ScenarioJoin Probability (p)Payout (b)Premium (a)Profit (x) = (a-b)xpx2p

No Fire99.8001%0540 540 538.92 291,017.09

One Fire0.0999%260000540 (259,460) (259.20) 67,252,172.11

Two Fire0.0001%520000540 (519,460) (0.52) 269,838.69

Summation 279.20 67,813,027.89

Expected Value (E)= Sum(XP)= 279.20

Variance (Var)= x2p-E2= 67,735,074.95

Standard Deviation = Sqrt(Var) 8,230

Answer E) From b & d, we can see that expected profits have increased drastically with increase in number of policies, though variance and S.D. of this expected profits remained similar